Boundary crossing probabilities for $(q,d)$-Slepian-processes

TL;DR

This paper derives an analytical formula for the boundary crossing probability of $(q,d)$-Slepian processes with piecewise affine boundaries, aiding in approximations for more complex boundary functions.

Contribution

It provides the first explicit analytical formula for boundary crossing probabilities of $(q,d)$-Slepian processes with piecewise affine boundaries, enabling better approximations.

Findings

Derived an explicit formula for boundary crossing probability

Applicable to piecewise affine boundary functions

Facilitates approximation for arbitrary boundaries

Abstract

For $0<q< d$ fixed let $W^{[q,d]}=(W^{[q,d]}_t)_{t\in {[q,d]}}$ be a $(q,d)$-Slepian-process defined as centered, stationary Gaussian process with continuous sample paths and covariance \begin{align*} C_{W^{[q,d]}}(s,s+t) = (1-\frac{t}{q})^+, \quad q\leq s\leq s+t\leq d. \end{align*} Note that \begin{align*} \frac{1}{\sqrt{q}}(B_t-B_{t-q})_{t\in [q,d]}, \end{align*} where $B_t$ is standard Brownian motion, is a $(q,d)$-Slepian-process. In this paper we prove an analytical formula for the boundary crossing probability $\mathbb{P}\left(W^{[q,d]}_t > g(t) \; \text{for some } t\in[q,d]\right)$, $q< d\leq 2q$, in the case $g$ is a piecewise affine function. This formula can be used as approximation for the boundary crossing probability of an arbitrary boundary by approximating the boundary function by piecewise affine functions.